Download Electronic Correlations in Transport through Coupled Quantum Dots V 82, N 17

VOLUME 82, NUMBER 17
PHYSICAL REVIEW LETTERS
26 APRIL 1999
Electronic Correlations in Transport through Coupled Quantum Dots
1
Antoine Georges1 and Yigal Meir 2
Laboratoire de Physique Théorique de l’Ecole Normale Supérieure,* 24 rue Lhomond, 75231 Paris Cedex 05, France
2
Department of Physics, Ben-Gurion University, Beer Sheva 84105, Israel
(Received 23 November 1998)
The conductance through two quantum dots in series is studied using general qualitative arguments
and quantitative slave-boson mean-field theory. It is demonstrated that measurements of the
conductance can explore the phase diagram of the two-impurity Anderson model. Competition between
the Kondo effect and the interdot magnetic exchange leads to a two-plateau structure in the conductance
as a function to the gate voltage and a two or three peak structure in the conductance versus interdot
tunneling. [S0031-9007(99)09017-1]
PACS numbers: 73.40.Gk, 72.15.Qm, 73.23.Hk
double-dot system can contain zero, one, or two electrons,
depending on the chemical potential: N ­ 0 for m ,
e2 , N ­ 1 for e2p , m , e1 , and N ­ 2 for m . e1 ,
with e6 ­ e0 6 De 2 1 t 2 . In the presence of a finite
antiferromagnetic spin exchange J between the dots, one
still has the three possibilities with e1 replaced by e1 2
3Jy4 [14]. In the Coulomb blockade regime there will be
two peaks in the conductance versus chemical potential
at the degeneracy points m ­ e6 . Alternatively, starting
from the N ­ 2 regime for t ­ 0, e1 will increase with
increasing t, until it crosses the chemical potential and the
The recent observation of the Kondo effect in transport through a quantum dot [1–3] opened a new path for
the investigations of strongly correlated electrons. Having confirmed earlier theoretical predictions [4,5] that a
quantum dot behaves as a magnetic impurity, these experiments also serve as a critical quantitative test for existing theories. In particular, unlike magnetic impurities
in metals which have physical properties determined by
the host metal and the impurity atom, the corresponding
parameters in the quantum dot case can be varied continuously, enabling, for example, a crossover from the Kondo
to the mixed-valence and the empty dot regimes in the
same sample [1,2].
The behavior of a lattice of magnetic impurities,
such as a heavy-fermion system, is characterized by the
competition between the Kondo effect and the magnetic
correlations between the impurities. An important step
towards the understanding of this problem was taken by
Jones and collaborators [6], who studied the two-impurity
problem. Their work demonstrated that this competition
leads to a second-order phase transition when particlehole symmetry applies. When this symmetry is broken,
this transition is replaced by a crossover [7–9]. In view of
the extensive experimental research on transport through
two dots in series [10,11], it is thus natural to try and
understand how this phase transition is manifested in the
double-dot system, both because such systems may have
important applications (such as a quantum-dot laser [12])
and because such a tunable system may reveal detailed
information on the corresponding phase diagram.
Transport through a double-dot system (see inset of
Fig. 1) has already received much theoretical attention,
in particular in the high temperature, Coulomb blockaded
regime [12,13]. In experiments the Coulomb charging
energy and the excitation energy are much larger than
temperature. Accordingly, only a single state on each
dot is important, and double occupancy of each dot can
be ignored. Denoting the energies of these states by
e1 ; e0 1 De and e2 ; e0 2 De, respectively, and the
tunneling amplitude between the dots by t, the isolated
FIG. 1. Plot of the conductance vs the tunneling between the
dots, t, obtained by slave-boson mean-field theory. Because
of the Kondo effect in the two-electron regime sN ­ 2d the
conductance has a peak at t ­ G. As t increases beyond t 1 ,
the Kondo effect is quenched, until the ground state contains a
single electron sN ­ 1d, leading to a different Kondo state and
an enhanced conductance. For finite antiferromagnetic coupling
J (finite a ­ G 2 yUTK0 ), the conductance peak is pushed to
smaller values of t and becomes narrower, as the singlet
formation destroys the Kondo state. In addition, a second
maximum in the conductance in the N ­ 2 regime emerges.
Inset: The double-dot system discussed in this paper.
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© 1999 The American Physical Society
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VOLUME 82, NUMBER 17
PHYSICAL REVIEW LETTERS
ground state will have a single electron in the double-dot
system. Again we expect a peak in the conductance at
t ­ t1 corresponding to e1 st1 d ­ m.
At low temperatures, in addition to a renormalization
of the above energy scales, e6 ! e 6 , and t1 ! t 1 , the
Kondo effect starts to play a significant role in the transport. The relevance of the Kondo effect in the doubledot system has been studied in [15,16]. Here, we focus
on the competition between the Kondo effect and antiferromagnetic exchange, and present detailed predictions
for the conductance. (Recently Andrei et al. [17] have investigated a very different realization of this competition,
which applies to coupled metallic islands, close to points
of charge degeneracy—i.e., near the Coulomb-blockade
peaks [18].) We find a rich phase diagram leading to interesting features in the conductance as a function of gate
voltage and intradot tunneling. As the corresponding energy and temperature scales are experimentally accessible,
these predictions are relevant to transport experiments in
double-dot systems.
To simplify notations, we assume in the following
e1 ­ e2 ­ e0 and VL ­ VR ­ V , where VL sVR d is the
coupling to the left (right) lead. Then the eigenstates of
the double-dot system are the even and odd states. As
even-odd symmetry is broken anyway by the tunneling
t, one can show that the above assumptions have little
effect on the underlying physics. Following Ref. [19], the
zero-temperature conductance per spin channel through
the system can be expressed in terms of the retarded
ret
svd
Green functions for the even and odd states Ge,o
e2 2
ret
ret
2
as g ­ h G jGe sv ­ 0d 2 Go sv ­ 0dj where G ;
prV 2 and r is the density of states in the leads at
the Fermi energy. Defining the corresponding scattering
phase shifts, da ; p 1 argGaret smd (such that da is in
the 0 p range), the conductance formula simplifies to
e2 2
g­
sin sde 2 do d .
(1)
h
The Friedel sum rule relates the total charge q per spin
channel, on the double-dot system, to these phase shifts:
q ­ sde 1 do dyp. There is, however, no individual
relation between da and the occupation of the corresponding state.
For m . e 1 (or alternatively t , t 1 ), there are N ­ 2
electrons in the system sq ­ 1d, and states with N ­ 0 or
N ­ 1 are high-energy states that can be eliminated from
the Hilbert space. The effective low-energy Hamiltonian
involves only spin degrees of freedom on the dots. It can
be cast into the form of a model of two Kondo impurities,
with Kondo couplings to the even and odd combinations
of the conduction electrons in the leads, an interimpurity magnetic exchange J ~ t 2 yU, and a potential scattering term in the leads Ve,o such that Ve 2 Vo ~ tV 2 y
se02 2 t 2 d.
Previous studies of the two-impurity Kondo problem
[mainly using Wilson numerical renormalization group
26 APRIL 1999
(NRG)] [6,8] have already yielded information on how
the phase shifts de , do behave as a function of the
couplings at T ­ 0 in this regime. Let us start with the
case J ­ 0. For t ­ 0 there is an even-odd symmetry,
and each channel has its own Kondo effect, leading to
de ­ do ­ py2, and naturally a zero conductance. As
tyG is increased, the difference de 2 do increases to
reach the value py4, at which point the conductance takes
its maximum possible value: e2 yh per spin channel. As
tyG is further increased, the Kondo effect is gradually
overcome by potential scattering and one reaches do . 0,
de . p in the large tyG limit (but still with t ø t 1 ). A
slave-boson mean-field theory (SBMFT) presented below
(cf. also Ref. [16]) yields in this regime sJ ­ 0, q . 1d
d ; de 2 do ­ 2 tan21 tyG, leading to
e2 4t 2 G 2
g­
,
(2)
h st 2 1 G 2 d2
which reaches its maximum value at t ­ G (solid curve
in Fig. 1). The Kondo temperature in this regime is
of order TK ­ c1 TK0 ec2 tyG , where the c’s are weakly
dependent on tyG and TK0 ; We2pje0 jyG is the single-dot
Kondo temperature. (The SBMFT yields c1 ­ cosdy2
and c2 ­ dy2.) The crucial content of that formula is
that the coupled-dot Kondo temperature can be much
larger than the single-dot Kondo temperature for small
J and large tyG. This has important consequences for the
observability of the effects described in this paper.
Let us now consider the effect of a finite J. For t ­ 0,
the effective two-impurity Kondo model has particlehole symmetry, and it is known from the work of [6]
that a phase transition exists at a critical value of the
coupling Jc yTK0 . 2.2. For J , Jc , the spin of each
dot undergoes a Kondo effect with the leads and de ­
do ­ py2. For J . Jc , the two spins are locked into
a singlet state and the Kondo effect does not apply,
yielding do ­ 0, de ­ p. The phase-shift difference d
jumps discontinuously from d ­ 0 for J , Jc to d ­ p
for J . Jc . (The conductance is, of course, zero for
all J since t ­ 0). Turning on a small value of tyG is
known to be a “relevant perturbation” on this critical point
1
(with dimension 2 , identical to that of J 2 Jc [7,8]) and
therefore smears the transition into a rapid crossover from
d ­ 0 to d ­ p. For J close to Jc , this smearing is
described by a crossover√scaling function:
!
d
sJ 2 Jc dyTK0
,
(3)
­f
p
tyG
with fsx ! 2`d ­ 0 and fsx ! 1`d ­ 1. As a result,
the conductance has a very sharp maximum as tyG
is increased for a fixed value of J close to Jc . For
J significantly larger than Jc , the conductance remains
very small with only a shallow maximum as tyG is
increased. For intermediate values of tyG and JyTK0 , a
quantitative calculation of d is needed in order to obtain
the conductance, using, e.g., NRG [6,8,20] or SBMFT.
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However, much can be said on semiquantitative level
by using existing knowledge on the two-impurity Kondo
problem. The phase shift d is an increasing function
of J, which starts at the value given above Eq. (2),
and increases until it saturates at d ­ p at a scale J p .
From the above estimate of TK the ratio J p yTK0 increases
exponentially with tyG. These considerations, and the
knowledge of the crossover around Jc [Eq. (3)], lead
to a qualitative contour plot of the conductance in the
sJyTK0 , tyGd parameter space, through the N . 2 regime,
as displayed in Fig. 2.
In practice, the exchange J is not an independent
parameter, but is a function of the interdot tunneling, J ,
t 2 yU. The contour plot above must thus be intersected by
a curve JyTK0 ­ astyGd2 , with a ; G 2 yUTK0 , in order to
follow the dependence of the conductance as a function of
tyG. Since TK0 is a very sensitive function of the energy
scales (such as e0 and G), the control parameter a can
be varied continuously over many orders of magnitude,
allowing an experimental investigation of most of the
phase diagram. Thus, as a function of t, the maximum
conductance e2 yh is reached for t . G with a peak width
Dt ~ G for small a, while the peakp is pushed down
to much lower transmission t . Gy a and becomes
very narrow Dt . Gya for large a. In addition, as the
saturation scale J p increases exponentially with t, one
may expect, for an intermediate a (middle broken curve
in Fig. 2), an additional peak in the conductance versus t
in the N ­ 2 regime. These results are indeed confirmed
by the SBMFT calculation (see Fig. 1).
As t is further increased st . t 1 d, the equilibrium
charge decreases to N ­ 1 sq ­ 1y2d. In this regime the
FIG. 2. Schematic contour map of the conductance in the
N . 2 regime. Thicker lines denote higher conductance, the
thickest one corresponding to g ­ e2 yh. The broken lines are
three physical contours (for different values of a ; G 2 yUTK0 )
along which J , t 2 yU.
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effective Hamiltonian is that of a single-impurity Kondo
problem in the even parity sector [15], leading to unitary
scattering de . py2. In the odd parity sector, we have
an almost empty resonant level with d0 . 0 [note that
sde 1 do dyp ­ q . 1y2 consistent with Friedel sum
rule]. Throughout this regime, we therefore expect the
zero-temperature conductance to be maximum g ­ e2 yh
and essentially independent of t. In this regime, the
interdot exchange J plays little role.
Similar interesting behavior is expected as a function
of gate voltage that controls the depth of the level energy
e0 with respect to the chemical potential (see Fig. 3). For
a very deep level the Kondo temperature is exponentially
s0d
small, and thus JyTK is large and quenches the Kondo
effect. As e0 increases, the Kondo temperature increases
and we enter the sN ­ 2d Kondo state, and a finite
conductance. This conductance remains constant (at zero
temperature) at a value smaller than e2 yh, determined by
the value of t, until e0 crosses the Fermi energy and a new
sN ­ 1d Kondo state is formed. There the conductance
is given by its maximum value, e2 yh per spin. As e0 is
further increased the double-dot system becomes empty
and the conductance drops to zero.
To substantiate these semiquantitative arguments we
have performed a quantitative calculation of the phase
shifts and the conductance using a slave-boson meanfield approximation. This method becomes exact as the
number of spin degrees of freedom goes to infinity, and
has been previously used in order to study the twoimpurity Anderson model in Ref. [7]. It was recently
applied in the present context in Ref. [16] but only in
FIG. 3. Plot of the conductance versus the level energy, as
obtained from SBMF T (for t ­ 2 and UyG ­ 104 ). The
conductance rises from a very small value (the singlet regime,
J ¿ TK ), to a t-dependent value (N ­ 2 Kondo regime,
J ø TK ), and then to g ­ e2 yh (N ­ 1 Kondo regime) before
dropping to zero for an empty dot.
VOLUME 82, NUMBER 17
PHYSICAL REVIEW LETTERS
the case J ­ 0. We have solved numerically the full
set of equations including the tunneling t and exchange
J, but we quote here only the simplified version of the
equations [7] that hold in the q ­ 1 regime (one electron
in the double-dot system per spin state, corresponding to
N ­ 2). For small enough values of JyTK0 , the phaseshift difference d is given by the solution of
!
√
J
2p dty2G
d
t
d
­ 0 .
(4)
sin
e
2 cos
d
2
G
2
TK
As J is increased beyond a critical coupling JcSB , d
reaches the value p: This is either a smooth transition for
t . 1yp or a first-order jump for t , 1yp (determined
by free-energy considerations). The existence of a phase
transition even for nonzero values of tyG is an artifact
of the SBMFT approximation: JcSB should actually be
interpreted as an estimate of the saturation scale J p
discussed above. This spurious transition does not affect
qualitatively the behavior of the conductance, except
when it becomes very small: there a strictly zero value
of g can be found (as evident on Fig. 1), whereas the
real system would have only a very small but finite g.
The SBMFT also provides a quantitative estimate of the
Kondo scale for the coupled-dot system in the q . 1
regime, as mentioned after Eq. (2).
In conclusion, we have demonstrated that measurements of the conductance through a double-dot system can
explore the phase diagram of the two-impurity Anderson
model. By changing the control parameter a ­ G 2 yUTK0
(which depends sensitively on the gate voltage), one can
make various cuts through the phase diagram (Fig. 2),
leading to nontrivial features in the conductance vs gate
voltage and interdot tunneling (Figs. 1 and 3). As the
relevant temperature scale can be much higher than the
single-dot Kondo temperature we believe that these predictions could be tested experimentally.
This work was supported in part by a grant from
the French-Israeli Scientific and Technical Cooperation
Program (Arc en Ciel-Keshet-1998). Work at BGU was
further supported by the The Israel Science FoundationCenters of Excellence Program.
*Unité propre du CNRS (UP 701) associée à l’ENS et à
l’Université Paris-Sud.
[1] D. Goldhaber-Gordon et al., Nature (London) 391, 156
(1998); D. Goldhaber-Gordon et al., Phys. Rev. Lett. 81,
5225 (1998).
[2] S. M. Cronenwett et al., Science 281, 540 (1998).
[3] For reviews on transport through quantum dots see, e.g.,
M. A. Kastner, Comments Condens. Matter Phys. 17, 349
(1996); R. Ashoori, Nature (London) 379, 413 (1996).
[4] L. I. Glazman and M. E. Raikh, Pis’ma Zh. Eksp. Teor.
Fiz. 47, 378 (1998) [JETP Lett. 47, 452 (1988)]; T. K. Ng
and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988).
26 APRIL 1999
[5] S. Hershfield, J. H. Davies, and J. W. Wilkins, Phys. Rev.
Lett. 67, 3720 (1991); Y. Meir, N. S. Wingreen, and P. A.
Lee, Phys. Rev. Lett. 70, 2601 (1993); N. S. Wingreen and
Y. Meir, Phys. Rev. B 49, 11 040 (1994).
[6] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev.
Lett. 61, 125 (1988); B. A. Jones and C. M. Varma, Phys.
Rev. B 40, 324 (1989); B. A. Jones, Physica (Amsterdam)
171B, 53 (1991).
[7] B. A. Jones, G. Kotliar, and A. J. Millis, Phys. Rev. B
39, 3415 (1989); A. J. Millis, G. Kotliar, and B. A. Jones,
in Field Theoretic Methods in Condensed Matter Physics,
edited by Z. Tesanovic (Addison-Wesley, Reading, MA,
1990), p. 159.
[8] O. Sakai and Y. Shimizu, J. Phys. Soc. Jpn. 61, 2333
(1992); 61, 2348 (1992).
[9] I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 68, 1046
(1992); I. Affleck, A. W. W. Ludwig, and B. A. Jones,
Phys. Rev. B 52, 9528 (1995).
[10] U. Sivan et al., Europhys. Lett. 25, 605 (1994); F. R.
Waugh et al., Phys. Rev. Lett. 75, 705 (1995); Phys.
Rev. B 53, 1413 (1996); N. C. Van der Waart et al.,
Phys. Rev. Lett. 74, 4702 (1995); F. Hofmann et al.,
Phys. Rev. B 51, 13 872 (1995); R. H. Blick et al., Phys.
Rev. B 53, 7899 (1996).
[11] For recent experimental studies of a double-dot system,
see, e.g., T. H. Oosterkam et al., Phys. Rev. Lett. 80, 4951
(1998); R. H. Blick et al., Phys. Rev. Lett. 81, 689 (1998).
[12] C. A. Stafford and N. S. Wingreen, Phys. Rev. Lett. 76,
1916 (1996).
[13] G. M. Bryant, Phys. Rev. B 44, 3064 (1991); 48, 8024
(1993); L. I. Glazman and V. Chandrasekhar, Europhys.
Lett. 19, 623 (1992); I. M. Ruzin et al., Phys. Rev. B 45,
13 469 (1992); G. Klimeck, G. Chen, and S. Datta, Phys.
Rev. B 50, 2316 (1994); G. Chen et al., Phys. Rev. B
50, 8035 (1994); J. J. Palacios and P. Hawrylak, Phys.
Rev. B 51, 1769 (1995); J. M. Golden and B. I. Halperin,
Phys. Rev. B 53, 3893 (1996); 54, 16 757 (1996); 56,
4716 (1997); K. A. Matveev, L. I. Glazman, and H. U.
Baranger, Phys. Rev. B 53, 1034 (1996); 54, 5637 (1996);
P. Pals and A. Mackinon, J. Phys. Condens. Matter 8, 3177
(1996); 8, 5401 (1996); R. Kotlyar and S. Das Sarma,
Phys. Rev. B 56, 13 235 (1997); C. A. Stafford, R. Kotlyar,
and S. Das Sarma, Phys. Rev. B 58, 7091 (1998);
Y. Asano, Phys. Rev.pB 58, 1414 (1998).
[14] However, for J . 8 De 2 1 t 2 y3 there will be a direct
jump from N ­ 0 to N ­ 2. Since in practice J stems
from superexchange between the dots, J , t 2 yU, the
latter case may not be experimentally relevant, as it
requires t , U.
[15] T. Ivanov, Europhys. Lett. 40, 183 (1997); T. Pohjola
et al., ibid. 40, 189 (1997).
[16] T. Aono, M. Eto, and K. Kawamura, J. Phys. Soc. Jpn. 67,
1860 (1998).
[17] N. Andrei, G. T. Zimanyi, and G. Schön, cond-mat /
9711098.
[18] K. Matveev, Sov. Phys. JETP 72, 892 (1991).
[19] Y. Meir and N. Wingreen, Phys. Rev. Lett. 68, 2512
(1992).
[20] W. Izumida, O. Sakai, and Y. Shimizu, cond-mat/
9805067.
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